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9. Electrostatic and magnetic effects

In any small force gravitational experiment electric charges on the test bodies must be absolutely avoided since they can easily produce forces far much bigger than the gravitational signal. The GG spacecraft does not contain free floating masses, and therefore no electrostatic charges will be able to build up inside it. Potential differences between the test masses can be avoided by coating them with a thin layer of the same conductive material. The PGB laboratory can be made of , or another highly conductive material, so as to work as a Faraday cage, shielding the experiment from external electric fields. Any small residual charge inside the spacecraft will only produce a constant effect on the output signal and can therefore be neglected. It is worth stressing that this is a very important advantage of GG as compared to STEP, where the test bodies are suspended by means of magnetic bearings and the problem of discharging them without producing unwanted perturbations is a serious one. The problem is even worse if the spacecraft orbit goes, as in the case of STEP, through the Van Allen belts and is subject to the bombardment of charged particles in the so-called South Atlantic Anomaly. The orbit of GG is equatorial and at a low enough altitude to avoid the South Atlantic Anomaly.

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Fig. 18. In an equatorial orbit the direction of the Earth's magnetic field is not more than about  away from the perpendicular to the orbit. Therefore its effects in the plane of the orbit will be reduced to about  times those of the full field. These effects are distinguishable from a violation of the equivalence principle because they change sign every half orbit.

Magnetic disturbances can be of two types: interactions of magnetized or magnetizable materials between themselves and interactions between these materials and the Earth's magnetic field. All magnetic and especially ferromagnetic materials inside the spacecraft should be avoided, i.e. magnets, electromagnets and electric motors. All electric currents should flow in shielded cables. For small controlled displacements we shall use piezoelectric actuators and active damping will be realized with electrostatic actuators. Any residual constant magnetization inside the spacecraft, as for the case of electrostatic charges, would produce only a DC effect (unless there is a relative motion at the spinning frequency). If the GG satellite is essentially non magnetic, its attitude is almost unaffected by the Earth's magnetic field  because on average over one orbit  is almost parallel to the spin axis. Along the satellite orbit the instantaneous orientation of  is no more than about  away from the spin axis of the satellite (Fig. 18 ). Thus, the component of  in the orbit plane can only produces a signal whose amplitude is reduced by a factor . If the interaction is between the proper test mass magnetization and the Earth's magnetic field, the frequency of this signal is , because the shape of the magnetic field rotates with the Earth. The signal also changes sign every half orbit of the satellite. When the interaction is between induced magnetization on the test mass and the Earth's magnetic field, the frequency is  because it depends on . We can argue in the same way for the component of  parallel to the spin axis. The interaction between magnetized test masses gives an  signal only if the interaction is between an induced magnetized test mass and a test mass with its own magnetization. A signal at this frequency could also arise if the test masses have their own magnetization and are subject to deformations (e.g. due to nonuniform thermal expansion) at the  frequency.

Let us now estimate the intensity of these perturbations. The most dangerous ones are those which act at (or close to) the spin/signal frequency. The largest among them is due to the interaction between the magnetic moment  (due to ferromagnetic impurities) of one test body and the magnetization induced on the other (with susceptibility ) by the magnetic field of the Earth. In the worst case hypotheses the resulting perturbing force which competes with the signal is (in MKS):

with  ( the permeability of vacuum),  the mutual distance and  the volume of body 1. For it to be smaller than the signal  it must be:

Since reasonable values for the susceptibility are , we need . From experimental data reported in textbooks we find that the magnetic moment of a cube of magnet of  size is about . Since this is in fact quite a large impurity, it appears that the inequality (Eq. 59 ) can be satisfied by the test bodies thus ruling out any need of reducing the magnetic field of the Earth inside the satellite. Another magnetic perturbation which acts at the spin/signal frequency is due to the interaction of the magnetic moment of one test body with the magnetic field of the Earth. Worst case hypotheses give:

hence we need , which can be easily satisfied. A DC magnetic perturbation comes from the interaction of the magnetic moments of two test bodies with one another:

For it to be smaller than the signal it must be , which, from the previous discussion, is a reasonable requirement for a DC effect. We complete the analysis by estimating the magnetic perturbations which contain  and therefore appear at a frequency close to twice the spin/signal frequency. One of these effects is due to the interaction of the magnetic moments induced on the test bodies with one another:

which requires  and is satisfied because . Another  effect comes from the interaction between B and the magnetization induced on a test body by B itself. The resulting perturbation force is:

and it is smaller than the signal provided that , which is surely the case.

In conclusion, as far as magnetic perturbations in the GG experiment are considered, we have demonstrated that the only one which sets a somewhat demanding requirement is due to the coupling of the magnetic field of the Earth with the magnetic moment of either test body due to residual ferromagnetic impurities. For this effect to be neglected we need the magnetic moment of the test bodies not to exceed a few , which appears to be feasible. On the contrary, in the torsion balance of the Eöt-Wash experiment (Su, 1992; Su et al., 1994) the magnetic field of the Earth near the balance was reduced by a total factor of  (partly with -metal shielding, partly with Helmotz coils). This seems to contradict our previous conclusion, especially if one considers that they have reached a sensitivity  while the GG target is . Indeed, it is not so and we are going to show why. The first important fact to bear in mind is that, despite its higher target sensitivity the GG expected force signal is  times larger than it is in Eöt-Wash, because of the bigger EP signal in space and the larger mass of the test bodies. In GG there are two test bodies of  each while in the Eöt-Wash torsion balance there are 4 masses of  each; the force signals are  and  respectively. Note that the force, not the acceleration, is relevant when dealing with nongravitational perturbations. Secondly, since the Eöt-Wash experiment is a torsion balance experiment it is sensitive to torques, hence also to the magnetic torque generated by the interaction of the magnetic field of the Earth with magnetic moments of the test bodies (due to residual ferromagnetic impurities). Indeed, it turns out that the magnetic moment of the tray on which the test bodies are positioned gives an even larger perturbation than the test masses themselves. For this torque to be smaller than that due to an EP violation it must be:

where  is the length of the arm. It must therefore be  (having used  as above, although the value used by Su et al. (1994) is actually ). From measurement of the torsion angle in absence of any shielding or coils the Eöt-Wash group finds that the residual magnetic moment of the tray (made of Al) is  (Su, 1992; Su et al., 1994), thus making a reduction of B by  crucial for the success of the experiment. This is achieved by means of a 3-layer -metal shielding for a factor  and of Helmotz coils for a factor 28. As for the Eöt-Wash test masses, the measured value of the residual magnetic moment is  while the requirement imposed by the magnetic torque is about ; with a factor  of reduction of the magnetic field of the Earth made necessary by the tray, this effect is no problem. The magnetic dampers, used to kill the swing and wobble modes so that the motor can provide a smooth rotation, will also benefit of the reduction of the magnetic field. In GG we have symmetric and concentric masses and the signal is a force, not a torque, thus we have nothing like the torque (Eq. 62 ); we do not have any motor or magnetic dampers either.

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       (Anna Nobili-